Rigid bodies with gyroscopic element
I'm fairly new to Blender and may not be much help; I'd leave a comment, but don't have that privilege yet :/ You'd think giving comment approval before answers... oh well.

Anyway, I think the cup would have to be attached to the cage, like in the real world.  The pins of the cage mesh would have to be inserted in the cup with holes in the cup axis to keep the cup from falling, not joined, like the real thing.

I would put the holes in the side of the cup with a reinforced thickness so it would have enough geometry that it would lay on top of the wire, and the wire wouldn't pass through this.  I've had to do that with objects standing on a plane, when the plane had no thickness even though it was a rigid body.  I wouldn't parent the cup to the frame either, the frame and the cup would have to affect each other, and parenting might/would prevent this.

This would allow the cup to 'hang' in the wire mesh with physics.  The other consideration would be based on the geometry of the cup, with a weighted cup, such that gravity would allow it to spin around the wire axles and hold the cup in the required position only stopped by the tab against the wire; in other words, bottom heavy.  That way, as in the real world, when it rotated to the top, the cup would stay properly oriented and it would contain the ball, and the tab would stop it, again, like the real thing.  I think you can also play with the center of gravity, if the cup is not perfectly shaped to hang like you want.  I think the cup would have to be a rigid body, as I'm sure the wire frame would be a rigid body, that way the cup would hang on the wire frame and not pass through it. You might also have to mess around with the material of the ball, so that it wouldn't have the same weight as a solid ball. My experience is they're more like ping pong balls, or small enough not to change the overall behavior of the cup.